Method for determining absolute spatial coordinates of at least one point on a reflecting surface

ABSTRACT

By localizing a location D on an optical axis, a virtual image is determined and compared with a measured virtual image. Using an iterative method, a location D can be shifted until the calculated and measured virtual images coincide and these data can then be used to determine an absolute spatial co-ordinate for a point on the surface. This method can be used, for example, in charting the surface of the cornea of an eye which has a reflecting film of tears.

BACKGROUND OF THE INVENTION

The invention relates to a method for determining absolute spatialcoordinates of at least one location on a reflecting surface.

Methods for taking measurements of reflecting surfaces are known. Thesemethods are based on the principle that a point with fixed spatialcoordinates is reflected by the surface and that the virtual image ismeasured with a video keratometer. Several such methods and devices areknown, for example, from U.S. Pat. No. 5,106,183, from U.S. Pat. No.5,110,200 and from the German Offenlegungsschrift 40 30 002.

These methods are suitable for determining relative data points, theyare, however, not suited for determining the exact spatial coordinatesof individual locations on a reflecting plane.

SUMMARY OF THE INVENTION

It is therefore an object of the invention to provide a method which canbe used to determine individual spatial coordinates of a reflectingplane. This object is solved by a method in which on the surface thereis reflected a location P₁ having a distance h to an optical axis, withthe line from location P₁ perpendicular to the optical axis defining thelocation O, an arbitrary location D at a distance d from location O onthe optical axis is defined as vertex of the fixation angle φ of P₁, thedistance I₁ of the virtual image of location P₁ to the optical axis iscalculated with D as vertex of the fixation angle, the distance I₂ ofthe virtual image of location P₁ to the optical axis is measured, d isvaried until the measured distance I₁ * coincides with the calculateddistance I₂ * at D*, and a spatial coordinate of the reflecting surfaceis calculated from the respective fixation angle φ* and the distance I₁*=I₂ *.

When individual spatial coordinates are measured with conventionalmethods, the problem arises that for measuring the exact dimensions of avirtual image, certain assumptions have to be made about the spatiallocation of a virtual image, which do not always correspond to theactual situation. For mathematical reasons, however, it is not possibleto measure the dimensions of the virtual image and the distance of thevirtual image from a fixed spatial coordinate at the same time.

The invention is based on the understanding that an arbitrary locationof the virtual image in space can be assumed as long as the measuredvalues are iteratively compared with the calculated values until theycoincide. Consequently, the iterative method makes it possible todetermine both the exact dimensions of the virtual image and thedistance from the virtual image to a fixed location.

It is advantageous if the location P₁ is a location of a Placido disc.By subdividing a Placido disc into segments, an arbitrarily large numberof reflecting locations can be measured which can be easily locatedagain based on their location and preferably their color in themeasurement.

Preferably, the invention is applied for measuring the cornea of theeye, for example for determining the shape of rigid contact lenses.

In order to limit the possible location of the location D on the opticalaxis, it is advantageous if the distance a between the location O andthe intersection S between the reflecting surface and the optical axisis determined by superimposing at least one centering object P₂.

In one embodiment of the invention, the distance I₁ * is calculatedaccording to the formula

    I.sub.1 =x tan α+h

with

    tan α=-h/(d-R.sub.s /2)

and ##EQU1##

The distance I₂ is preferably measured with a video keratometer.

In another embodiment of the invention, the entire shape of a reflectingsurface is determined by determining the spatial coordinates of aplurality of locations and subsequently interpolating therebetween.

BRIEF DESCRIPTION OF PREFERRED EMBODIMENTS

An embodiment of the method of the invention is depicted in the drawingsand is described in further detail hereinafter.

It is shown in:

FIG. 1 a schematic representation of the geometrical arrangement, and

FIG. 2 an example of a calibration function.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

For measuring the cornea of an eye, the reflecting lachrymal film of theeye is used in which arbitrary locations, for example the point P₁ inFIG. 1, are reflected. If a considerable number of locations on thesurface are determined, then the shape of the entire surface of thecornea can be determined by interpolation.

For determining the spatial coordinates, a Placido disc with rings ofdifferent colors is reflected onto the lachrymal film of the eye. First,a location P₁ at a distance h to the optical axis 1, is observed on thereflecting surface with a video keratometer.

On the optical axis 1 there is located a zero point O which is obtainedby constructing a line from location P₁ perpendicular to the opticalaxis 1. In addition, an arbitrary location D at a distance d from thezero location 0 is defined on the optical axis 1 which is considered asfixation angle φ of P₁.

Subsequently, the distance I₁ which is the distance between the virtualimage of P₁ and the optical axis, is calculated. The starting point isthe fundamental relationship

    I.sub.1 =x tan α+h

with

    tan α=-h/(d-R.sub.s /2)

and ##EQU2##

By fixing the location D on the optical axis, all values for the aboveequations are known. The value of the sagittal radius RS is determinedfrom a calibration function, an example of which is depicted in FIG. 2.For determining the function RS=f(i), designated spheres are measuredwith a video keratometer during a calibration.

In addition, the distance I₂ of the virtual image of location P₁ to theoptical axis is measured; this measurement, too, is based on thesagittal radius previously determined through calibration.

Since point D can be any location on the optical axis, it has to beassumed that the calculated value I₁ is different from the measuredvalue I₂. If D is too far away from the zero point O, then thecalculated distance I₁ is smaller than the measured distance I₂ and thedistance d is subsequently decreased. The same calculation is repeatedwith the newly obtained location D, and the distances I₁ and I₂ arecompared at the end. Convergence of the location D with the location D*,where I₁ *=I₂ *, can, for example, be achieved by halving the interval.The accuracy of the method is determined by the resolution of the videotopometer, and when this accuracy has been achieved, the iteration isfinally terminated and the spatial coordinate of the location of thereflecting surface is calculated from the fixation angle and thedistance I₁ *=I₂ *.

In this manner, a large number of spatial coordinates can be determinedwherein different locations P are reflected on the lachrymal film of theeye and wherein the method is executed for each one of these locationson the surface. The result is a data set describing the shape of thesurface. The regions between the calculated locations are determinedthrough extrapolation.

In order to limit the range where the location D is located on theoptical axis 1, it is meaningful to determine first the distance abetween the point O and the intersection S between the reflectingsurface and the optical axis by superimposing at least one centeringobject. Here two laser beams forming an angle therebetween can bepointed at the surface in such a way that their reflected locationsintersect, whereby the distance can be calculated backwards from theangle. Alternately, only a single laser beam can be pointed at thesurface in such a way that the distance between the image and theoptical axis becomes zero, whereby the distance can be calculated fromthis geometrical relationship. The latter method is described in detailin WO 94/16611.

By determining the location S on the vertex of the test object, theactual contours can be delineated in conjunction with the spatialcoordinates determined with the method of the invention. Any distancebetween the system and the surface of the test object can thus bedetermined by combining the iterative method with the determination ofthe distance a, which is called z-correlation.

In addition, a ray tracing model can be constructed from these data anda geometrical and/or quantum-mechanical wave analysis (Snell's law). Byassuming a known reflective body (parallel rays, lattice design, etc.),the image distortion from the surface of the test object can bedetermined. This can be performed both axially (with reference to theoptical axis) and "instantaneously" through a determination of the truelocal radius, without other assumptions or limitations.

After different spatial coordinates of the reflecting surface have beendetermined, the surface can be described by an n^(th) -order polynomial,a Zernike polynomial or a spline function. Furthermore, the simulationof the sagittal-radii-method can be performed based on the determineddata, wherein the data and coefficients, respectively, obtained by thesimulated sagittal-radii-method can be represented by a relative orabsolute color scale.

In addition, it is proposed that a comparison between curvatures permitsthe determination of concave and convex structures and that the positive(convex) and negative (concave) radii can be represented by a relativeor absolute color scale.

What is claimed is:
 1. Method for determining absolute spatialcoordinates of at least one location on a reflecting surfacewherein onthe surface there is reflected a location P₁ having a distance h to anoptical axis 1, with the line from location P₁ perpendicular to theoptical axis 1 defining the location O, an arbitrary location D at adistance d from location O on the optical axis is defined as vertex ofthe fixation angle φ of P₁, the distance I₁ of the virtual image oflocation P₁ to the optical axis is calculated with D as vertex of thefixation angle, the distance I₂ of the virtual image of location P₁ tothe optical axis is measured, d is varied until the measured distanceI₁ * coincides with the calculated distance I₂ * at D*, and a spatialcoordinate of the reflecting surface is calculated from the respectivefixation angle φ* and the distance I₁ *=I₂ *.
 2. Method according toclaim 1, wherein the location P₁ is a location of a luminous ring of aPlacido disc.
 3. Method according to claim 1 wherein, characterized inthat the surface is the reflecting lachrymal film of an eye.
 4. Methodaccording to claim 1 wherein, characterized in that the distance abetween the location O and the intersection S between the reflectingsurface and the optical axis is determined by superimposing at least onecentering object P₂.
 5. Method according to claim 1, wherein thedistance I₁ * is determined according to the formula

    I.sub.1 =x tan α+h

with

    tan α=-h/(d-R.sub.s /2)

and ##EQU3## wherein I₁ is the distance of the virtual image of thelocation P₁ to the optical axis 1, h is the distance of the locations P₁to the optical axis 1, d is the distance of the location D from locationO, and R_(s) is the sagittal radius.
 6. Method according to claim 1wherein, characterized in that the distance I₂ is measured with a videokeratometer.
 7. Method according to claim 1 wherein, characterized inthat a reflecting surface is determined by determining a plurality ofspatial coordinates and by an interpolation.